WASHINGTON — Saying the opioid crisis requires bold measures, the state of Arizona filed an audacious lawsuit in the Supreme Court on Wednesday asking the justices to order members of the Sackler family, which owns Purdue Pharma, to return what the state said were billions of dollars looted from the company.

“We want the Supreme Court to make sure that we hold accountable those individuals who are responsible for this epidemic,” said Mark Brnovich, Arizona’s attorney general. “We allege that the Sacklers have siphoned billions of dollars from Purdue in recent years. They did this while knowing the company was facing massive financial liabilities.”

Lawsuits making similar claims have been brought in several state courts. What distinguishes the new suit is that it was filed directly in the Supreme Court, which almost never hears cases until after lower courts have considered them.

He added that the urgency of the crisis warranted the unusual move. “We don’t have time for this to take years to wind through the courts,” he said. “The Supreme Court has jurisdiction, and we think they have to act.”

A spokesman for members of the Sackler family called the accusations in the new lawsuit “inconsistent with the factual record” and said the family “will vigorously defend against them.” A lawyer for Purdue, which makes the opioid OxyContin, did not respond to requests for comment.

In papers filed in the Supreme Court on Wednesday, lawyers for Arizona wrote that the opioid crisis had contributed to hundreds of thousands of deaths in the last two decades and cost the United States economy more than $78 billion annually. Over the years, the lawyers wrote, Purdue earned more than $30 billion from sales of OxyContin.

Between 2008 and 2016, Purdue transferred more than $4 billion to the Sacklers, according to the new lawsuit. It is commonplace and lawful, of course, for a company’s owners to withdraw profits. But the suit contends that the transfers were intended to frustrate efforts by victims of the opioid crisis to obtain compensation.

“These transfers,” the suit said, “all took place at times when company officials, including the Sacklers, were keenly aware that Purdue was facing massive financial liabilities and that these transfers could prevent it from satisfying eventual judgments.”

The Sacklers are one of the richest families in the United States, and its members have made significant donations to museums and other cultural institutions. Several museums have announced that they will no longer take money from the family, and the Louvre Museum in Paris recently said that it had removed the Sackler name from its Sackler Wing of Oriental Antiquities.

The Constitution gives the Supreme Court “original jurisdiction” to hear disputes “in which a state shall be party.” In such cases, the Supreme Court acts much like a trial court, appointing a special master to hear evidence and issue recommendations.

Though the Constitution seems to require the court to hear cases brought by states, the court has ruled that it has discretion to turn them down and often does. When the court does exercise its original jurisdiction, it is usually to adjudicate disputes between two states over issues like water rights.

In 2016, the justices turned down a request from Nebraska and Oklahoma to file a challenge to Colorado’s legalization of recreational marijuana. The states said the Colorado law had spillover effects, taxing neighboring states’ criminal justice systems and hurting the health of their residents.

Justice Clarence Thomas, joined by Justice Samuel A. Alito Jr., dissented, saying that the case presented a substantial question and that the court was required to hear it.

“Federal law does not, on its face, give this court discretion to decline to decide cases within its original jurisdiction,” Justice Thomas wrote.

Drawing on that dissent, Arizona asked the Supreme Court to overrule decisions that allowed it to turn down cases filed by states directly in the court. Failing that, Arizona asked the court to allow it to sue in the Supreme Court as a matter of discretion.

“Absent resolution in a single forum,” the state’s lawyers wrote, “these disputes will be fought over and over in nearly every state in the nation. This is likely to take years, lead to inconsistent judgments and create an inequitable distribution of money damages.”

“The urgency is a big deal here.” he said. “It’s very important that we get this resolved expeditiously, and that’s one of the key reasons why the Supreme Court is the right place to do this and to do this now.”

Mr. Consovoy is also representing President Trump in legal clashes with House Democrats seeking his tax returns and business records.

Arizona’s lawsuit is based on the Uniform Fraudulent Transfer Act, which has been adopted in most states and forbids companies from making some kinds of transfers of assets when they are insolvent or likely to run out of money to pay creditors. The Supreme Court does not ordinarily hear cases based on asserted violations of state laws.

The unusual Supreme Court filing may reflect jockeying among states and local governments for a leading role in settlement negotiations. Vast sums may be at stake.

In March, the company and the family agreed to pay $270 million in connection with the settlement of a suit brought by the State of Oklahoma. In October, Judge Dan Aaron Polster of the Federal District Court in Cleveland is scheduled to preside over trials in a test case even as he seeks to resolve some 2,000 federal lawsuits brought by cities, counties and Native American tribes against key players in the opioid crisis.

A version of this article appears in print on , Section A, Page 19 of the New York edition with the headline: Arizona Sues Sacklers Over Opioid Crisis, Starting in the Supreme Court. Order Reprints | Today’s Paper | Subscribe